Tidal effects in the total flux and waveform in massless scalar-tensor theories to, respectively, relative 2PN and 1.5PN orders

This paper utilizes the post-Newtonian multipolar-post-Minkowskian formalism within scalar-tensor theories to compute tidal corrections to the total energy flux and waveform phasing of neutron star binaries up to 2PN and 1.5PN orders, respectively, while accounting for gravitational, scalar, and mixed tidal deformability effects essential for interpreting future high-precision gravitational wave data.

The Gist

Imagine two massive dancers, neutron stars, locked in a tight, spiraling embrace as they hurtle toward each other. As they spin faster and faster, they don't just move through space; they ripple the very fabric of the universe, creating gravitational waves. These waves are like the sound of a cosmic drumbeat, carrying information about the dance to our detectors on Earth.

For decades, physicists have modeled this dance assuming the stars are perfect, rigid spheres—like billiard balls. But in reality, neutron stars are more like giant, squishy water balloons made of ultra-dense matter. As they get close, the gravity of one star pulls on the other, stretching and squeezing it. This is called tidal deformation.

This paper is a massive upgrade to the "dance manual" used by scientists. Here is what the authors, Eve Dones and Laura Bernard, have done, explained simply:

1. The New "Gravity" Rules (Scalar-Tensor Theories)

Standard physics (General Relativity) says gravity is just the curvature of space. But this paper explores Scalar-Tensor theories, which suggest there might be a second, invisible "force field" (a scalar field) acting alongside gravity.

• The Analogy: Imagine the dance floor isn't just a flat surface (gravity); it's also covered in a thick, invisible gel (the scalar field). When the dancers move, they don't just push the floor; they also push through the gel. This gel changes how the stars feel each other's pull and how they wiggle.

2. The "Squishiness" of the Stars

The authors calculated exactly how these stars get distorted by two things:

1. The other star's gravity (like a hand squeezing a stress ball).
2. The invisible scalar field (like the gel pushing back).

They found that in this new theory, the stars don't just have one kind of "squishiness." They have three different types:

• Tensorial: The standard stretching you'd expect (like pulling taffy).
• Scalar: A unique stretching caused by the invisible gel field.
• Mixed: A weird combination of both.

3. The "Soundtrack" of the Dance (Waveforms)

When the stars dance, they emit a "soundtrack" (the waveform). To predict what our detectors (like LISA or the Einstein Telescope) will hear, we need to know the exact notes and rhythm.

• The Problem: Previous manuals only knew the first few notes of the song. They missed the subtle harmonics caused by the stars getting squished.
• The Solution: The authors calculated the "soundtrack" with much higher precision. They went from knowing the main melody to hearing the complex harmonics.
  • They calculated the energy loss (how fast the dance is speeding up) with extreme accuracy.
  • They calculated the phase (the exact timing of the beats) to ensure we don't lose the signal in the noise.

4. The "Echo" (Memory)

One of the coolest things they added is the memory effect.

• The Analogy: Imagine a drum. When you hit it, it makes a sound. But if you hit it hard enough, the drum skin doesn't just stop vibrating; it stays slightly stretched out afterward. The universe is similar. The passage of these gravitational waves leaves a permanent "scar" or "echo" in spacetime.
• The authors calculated this "echo" for the first time in this specific theory, including how the invisible scalar field changes the shape of that echo.

Why Does This Matter?

Think of gravitational wave detectors as incredibly sensitive ears. In the future, they will hear the universe with crystal clarity.

• If we use an old, inaccurate "dance manual" (General Relativity only), we might hear a sound and think, "That's just a normal star."
• But if the stars are actually "squishy" and interacting with the invisible "gel" (Scalar-Tensor theory), the sound will be slightly different.
• The Goal: By creating this ultra-precise manual, the authors give scientists the tools to tell the difference between a standard star and a star in a modified gravity theory. If we detect a "squishy" signature that matches their calculations, it would be a smoking gun proving that Einstein's theory of gravity isn't the whole story—that there is a new, invisible force at play.

In short: They took the complex math of how two super-dense stars stretch and squish each other in a universe with an extra invisible force, and they turned it into a precise instruction manual. This manual will help us decode the future messages from the cosmos and potentially discover new laws of physics.

Technical Summary

1. Problem Statement

The paper addresses the modeling of gravitational waves (GWs) emitted by inspiralling compact binary systems (specifically neutron stars) within massless scalar-tensor (ST) theories of gravity. While General Relativity (GR) has been extensively modeled, alternative theories like ST theories introduce a scalar field that couples to matter, leading to distinct phenomena such as dynamical scalarization and dipolar radiation.

The specific challenge tackled is the accurate modeling of tidal deformations in these theories. Neutron stars in binary systems are deformed by their companion's gravitational field and the scalar field. These deformations depend on the star's internal structure and composition. To interpret future high-precision GW data (e.g., from LISA or the Einstein Telescope) and distinguish between modified gravity signatures and standard matter properties, it is necessary to compute tidal corrections to the GW energy flux and waveform at high post-Newtonian (PN) orders.

Previous work had established the leading-order (LO) dipolar tidal effects. This paper aims to extend these calculations to the Next-to-Next-to-Leading Order (NNLO) for the energy flux and N1.5LO (Next-to-Next-to-Leading Order in the waveform, corresponding to relative 1.5PN) for the waveform amplitude, accounting for three independent types of tidal deformability: scalar, tensorial, and mixed scalar-tensorial.

2. Methodology

The authors employ a rigorous analytical framework combining several advanced techniques:

• Formalism: They utilize the Post-Newtonian Multipolar-Post-Minkowskian (PN-MPM) formalism, adapted specifically for scalar-tensor theories. This allows for a systematic expansion in small velocities ($v/c$) and weak gravitational fields.
• Action and Field Equations: The study starts with the Einstein-frame action where the scalar field is minimally coupled to the metric. The matter sector is treated using an Effective Field Theory (EFT) approach, adding non-minimal worldline couplings to the point-particle action to describe tidal effects. The tidal action includes terms for scalar-type ($\lambda, \mu$), tensorial-type ($c$), and mixed scalar-tensorial ($\nu$) deformabilities.
• Adiabatic Approximation: The calculation assumes an adiabatic inspiral, where the orbital evolution is slow compared to the orbital period.
• Radiative Sector:
  • The authors compute the total energy flux (gravitational + scalar) by deriving the radiative multipole moments (mass-type $U_L$, current-type $V_L$, and scalar-type $U^s_L$).
  • They account for non-linear memory effects (specifically $m=0$ modes) which, although formally appearing at higher PN orders, contribute at lower orders in the waveform due to PN order promotions.
  • They handle tail effects (hereditary terms) arising from the scattering of waves off the background curvature, which are crucial for the scalar dipole moment.
• Coordinate Systems: The derivation involves transforming from harmonic coordinates (used for the near-zone dynamics) to radiative coordinates (used for the far-zone waveform) to remove logarithmic terms and define the Bondi-type expansion.
• Circular Orbits: The results are specialized to quasi-circular orbits, expressing all quantities in terms of the gauge-invariant PN parameter $x = (\tilde{G}\alpha m \omega / c^3)^{2/3}$.

3. Key Contributions

This work represents a significant step forward in the precision of GW modeling for alternative gravity theories:

1. NNLO Tidal Corrections to Flux: The authors compute the tidal corrections to the total energy flux (gravitational and scalar) up to NNLO (relative 2PN order beyond the leading dipole). This is the first time the full set of three tidal deformability types (scalar, tensorial, mixed) has been included in the radiative sector at this accuracy.
2. N1.5LO Waveform Amplitudes: They derive the full gravitational and scalar waveform amplitude modes up to N1.5LO (relative 1.5PN). This includes the dominant modes and the memory modes ($m=0$) computed up to NLO, a novel contribution for ST theories.
3. Dipolar-Driven Regime Focus: The analysis focuses on the "dipolar-driven" (DD) regime, where the scalar dipole radiation dominates over the quadrupole, which is typical for ST theories with scalarized neutron stars.
4. Consistency Checks: The results are cross-checked against previous leading-order calculations and compared with an independent study (Ref. [26]) using a different conformal frame convention, establishing a translation map between Jordan-frame and Einstein-frame tidal parameters.

4. Key Results

The paper provides explicit analytical expressions for:

• Orbital Dynamics: The orbital frequency and conserved energy/angular momentum including NNLO tidal corrections.
• Energy Flux:
  • Gravitational Flux ($F_g$): Includes contributions from scalar, tensorial, and mixed tidal deformabilities at NNLO.
  • Scalar Flux ($F_s$): Includes similar tidal contributions.
  • The fluxes are expressed in terms of the PN parameter $x$ and the ST theory parameters (sensitivities $s_A$, coupling functions, etc.).
• Orbital Phase:
  • Time Domain: An analytical expression for the orbital phase $\phi(t)$ up to NNLO.
  • Frequency Domain (SPA): The phase $\Phi(f)$ within the Stationary Phase Approximation, essential for data analysis. The phase includes a characteristic $\log(v)$ term arising from tail effects.
• Waveform Modes:
  • Gravitational Modes ($h_{\ell m}$): Explicit tidal corrections for modes $(2,2), (2,1), (3,3), (3,2), (3,1), (4,4), (4,2)$ and the memory modes $(2,0), (4,0)$.
  • Scalar Modes ($\psi_{\ell m}$): Explicit tidal corrections for modes $(0,0), (1,1), (2,2), (3,3), (3,1), (4,4), (4,2)$.
  • Key Finding: At the N1.5LO accuracy of the waveform, only dipolar scalar tidal effects contribute. Quadrupolar scalar, tensorial, and mixed effects only enter at the NNLO (requiring a 2.5PN waveform calculation).

5. Significance and Implications

• Precision for Future Detectors: As GW detectors move toward the "third generation" (Einstein Telescope, Cosmic Explorer) and space-based (LISA), the signal-to-noise ratio will increase, making it possible to detect subtle deviations from GR. Accurate templates including NNLO tidal effects are essential to avoid biases in parameter estimation.
• Disentangling Gravity vs. Matter: The paper highlights that in ST theories, tidal effects are not just about the equation of state (EOS) of neutron stars but also about the scalar field coupling. Without these high-order corrections, one might misinterpret a modified gravity signature as a specific EOS or vice versa.
• Theoretical Completeness: By deriving the full set of tidal deformabilities (scalar, tensor, mixed) and memory effects, the authors provide a complete theoretical framework for testing scalar-tensor gravity. The inclusion of memory modes ($m=0$) is particularly important for capturing the non-linear nature of gravity in these theories.
• Future Directions: The authors note that to capture quadrupolar tidal effects in the waveform (not just the flux), computations must be extended to NNLO for the waveform itself (2.5PN relative order). They also suggest future work should include dynamical tides, spin effects, and extensions to theories with higher-curvature corrections (e.g., Einstein-scalar-Gauss-Bonnet).

In summary, this paper provides the necessary high-precision analytical tools to model tidal interactions in scalar-tensor theories, bridging the gap between theoretical predictions and the high-precision observational capabilities of next-generation gravitational wave astronomy.